Mark Chang

To the author, a paradox is “a statement or phenomenon that seems contradictory but in reality expresses a possible truth.” The book begins with a candy store of multi-disciplinary paradoxes, likely known to the sophisticated reader, but a pleasure to find in aggregate. Since “paradox” means literally, “alongside the faith,” the reader might want some immediate intuitive unravelings. But in fact the author briskly leads us deeper and deeper along the paradox trail into probability, statistics, and science.

Where probability is involved, we are immediately out-foxed by numbers. The drunken man fishing for his key on a key ring has the best probability of finding the correct key first. Probability theory insists that if the second key is the correct one, the first key must be wrong, which is an additional expectation. Surprise comes from a pair of games, in both of we expect to lose, but jumping between games offers an expected win. Or we may choose the “best” in a sequence of n candidates this way: wait on the first \(\frac{\varepsilon}{100}\)%, and then accept the next one found to be better.

But what does probability mean when, after all, P (plane crashes) = ½? As the paradoxes are more formally presented, the explanations do not become more obvious. What is meant by “random”? No formal definition exists. And what exactly do we know and when do we know it? Consider Intransitive Dice, or the Three Hat Paradox. In Lewis Carroll’s Urn, the act of adding an extra ball interferes with the time frame. And in the Two Envelope Paradox, it is eternally preferable to choose the “other.”

The second half of the book brings paradox to its most fertile soil — statistics and science — and the language becomes more technical. Although all technical concepts are breezily introduced, a strong first year course in probability and statistics is probably necessary for the reader. Presumably within the author’s research field of biometrics, the paradoxes now become dilemmas reflecting the insufficiency of statistics to accommodate data and human intuition in important decisions. Frequentism and Bayesianism disagree in the Lindley Paradox. The “black hole of scientific discovery” is “multiplicity” or aggregate error control when multiple tests are conducted. In Simpson’s Paradox applied to sex bias at Berkeley, should we accept the aggregate or the partitioned data?

Vectoring toward science, the author unexpectedly likens Occam’s Razor to “yin-yang.” In fact, there are enough Chinese references to suggest that the author, rather than following the Western style of driving home a point, uses a rhetorical style in which argument spirals outward. By the end, we are almost out of orbit with epistemological problems: free will, perception of time, limits of language, and what we can know. The Omnipresent Predictor, going back to kill your grandfather, Gödel’s Theorem, EPR, Traveling Twins, the Halting Problem — these are all here and much more. Simplicity in this sort of boggy terrain means the reader may need alternate sources for clarification. And the editing falls off. The concluding design for Artificial Intelligence architecture is out of range for this dilettantish reviewer. But the brisk pace makes the pages turn easily and the book is fun, especially when one looks up from the book, as one will want to do, in a roomful of kibitzers.

Sandra Z. Keith is professor emerita at St. Cloud State University, MN.

The Joy of Paradoxes: A Random Walk
Introduction to Paradox
Applications of Paradoxes
Mathematical Paradox
Probabilistic and Statistical Paradoxes